A lot of people contrive US citizenship for their children in order that they may gain the advantages of such citizenship. Mexican anchor babies are one example. A mom having a kid in Kenya would plausibly fake a US birth for her child too, given the chance.
She wouldn't need to go through that. The child would be a US citizen because the mother was, though not natural born.
Plus his father, if it's who he says it was, was not a US citizen therefore anyway you look at it, zer0 is ineligible.